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My old linear-algebra teacher, whom I can no longer ask, wrote on a black board an exclamation mark over the binary symbol of a logical formula, the main symbol of which is that binary symbol, to say that the formula is yet unproven but is about to be proven next. For example:

We now turn to $\varphi \stackrel{!}{\Leftrightarrow} \psi$:

$\Leftarrow$”: … [here comes a proof of $\varphi \Leftarrow \psi$] …

$\Rightarrow$”: … [here comes a proof of $\varphi \Rightarrow \psi$] …

Who was the/a first to introduce this kind of convention and with which meaning exactly? References (ideally, more substantial than „L. Euler, Private communication“ or “Emmy Noether, lectures on algebra in 1934 in Bryn Mawr” :-). )?

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    $\begingroup$ You sure anyone else ever did that? $\endgroup$ Sep 23 at 13:15
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    $\begingroup$ I wouldn't hold my breath. These homegrown didactic notations, for blackboard only, pop up here and there all the time and are near impossible to keep track of because they never make it into publications. For what it is worth, a question mark there makes more sense than an exclamation mark. $\endgroup$
    – Conifold
    Sep 24 at 7:40
  • $\begingroup$ I would put the question mark there. I do not think that it is standard or widely used. $\endgroup$
    – markvs
    Sep 25 at 23:35
  • $\begingroup$ I have done that, usually above an equality sign rather than a logical equivalence, to indicate the equality is surprising. $\endgroup$
    – KCd
    Sep 26 at 13:02
  • $\begingroup$ @CarlWitthoft At my Alma Mater, I think yes, though it has been long ago. $\endgroup$ Sep 28 at 23:21

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